[Proof of Lemma REF ] Note that any \(p\) -orbit \(O_{p,\beta _p}\)  intersects with any \(q\) -orbit \(O_{q,\beta _q}\)  at exactly one element \(\zeta ^{f(\beta _p,\beta _q)}\) , by using Chinese Remainder Theorem [1]}. Hence a disjoint union of orbits must by a union of orbits in the same length. So either \(p|\#(A)\)  or \(q|\#(A)\) . Thus \(\#(A)\ne k\)  since \(\gcd (k,pq)=\gcd (k,2k+1)=1\) .
